Singular 2-0-4 Manual: C.3 Syzygies and resolutions

C.3 Syzygies and resolutions

Syzygies

Let be a quotient of $\hbox{Loc}_< K[\underline{x}]$ and let be a submodule of . Then the module of syzygies (or 1st syzygy module, module of relations) of , syz(), is defined to be the kernel of the map $R^s \rightarrow R^r,\; \sum_{i=1}^s w_ie_i \mapsto \sum_{i=1}^s w_ig_i$ .

The k-th syzygy module is defined inductively to be the module of syzygies of the -st syzygy module.

Note, that the syzygy modules of depend on a choice of generators . But one can show that they depend on uniquely up to direct summands.

Example:

  ring R= 0,(u,v,x,y,z),dp;
  ideal i=ux, vx, uy, vy;
  print(syz(i));
→ -y,0, -v,0, 
→ 0, -y,u, 0, 
→ x, 0, 0, -v,
→ 0, x, 0, u

Free resolutions

Let $I=(g_1,...,g_s)\subseteq R^r$ and . A free resolution of is a long exact sequence

$\begin{displaymath}...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F... ...1}\over{\longrightarrow} F_0\longrightarrow M\longrightarrow 0,\end{displaymath}$

where the columns of the matrix

generate

. Note, that resolutions need not to be finite (i.e., of finite length). The Hilbert Syzygy Theorem states, that for $R = \hbox{Loc}_< K[\underline{x}]$ there exists a ("minimal") resolution of length not exceeding the number of variables.

Example:

  ring R= 0,(u,v,x,y,z),dp;
  ideal I = ux, vx, uy, vy;
  resolution resI = mres(I,0); resI;
→  1      4      4      1      
→ R <--  R <--  R <--  R
→ 
→ 0      1      2      3      
→ 
  // The matrix A_1 is given by
  print(matrix(resI[1]));
→ vy,uy,vx,ux
  // We see that the columns of A_1 generate I.
  // The matrix A_2 is given by
  print(matrix(resI[3]));
→ u, 
→ -v,
→ -x,
→ y

Betti numbers and regularity

Let be a graded ring (e.g., $R = \hbox{Loc}_< K[\underline{x}]$ ) and let $I \subset R^r$ be a graded submodule. Let

$\begin{displaymath} R^r = \bigoplus_a R\cdot e_{a,0} \buildrel A_1 \over \longl... ...ts \longleftarrow \bigoplus_a R\cdot e_{a,n} \longleftarrow 0 \end{displaymath}$

be a minimal free resolution of

considered with homogeneous maps of degree 0. Then the graded Betti number $b_{i,j}$ of

is the minimal number of generators $e_{a,j}$ in degree

of the

-th syzygy module of

(i.e., the

-st syzygy module of

). Note, that by definition the

-th syzygy module of

and the 1st syzygy module of

The regularity of is the smallest integer

such that

$\begin{displaymath} \hbox{deg}(e_{a,j}) \le s+j-1 \quad \hbox{for all $j$.} \end{displaymath}$

Example:

  ring R= 0,(u,v,x,y,z),dp;
  ideal I = ux, vx, uy, vy;
  resolution resI = mres(I,0); resI;
→  1      4      4      1      
→ R <--  R <--  R <--  R
→ 
→ 0      1      2      3      
→ 
  // the betti number:
  print(betti(resI), "betti");
→            0     1     2     3
→ ------------------------------
→     0:     1     -     -     -
→     1:     -     4     4     1
→ ------------------------------
→ total:     1     4     4     1
  // the regularity:
  regularity(resI);
→ 2

User manual for Singular version 2-0-4, October 2002, generated by texi2html.